4351
J. Am. Chem. SOC.1991,113,4351-4353 Scheme I
Table I. Equilibrium and Rate Constants for Scheme I Obtained from a Nonlinear-Least-Squares Fit to the Experimental Data at 6 OC pH 5.8 pH 7.0 7.8 f 1.1 6.7 i 0.5 ki (s-I) 0.5 0.03 0.4 0.03 k-, (PI) 16.3f 1.8 8.8 f 0.8 kz (s-') 0.6 f 0.05 0.4 f 0.05 k-2 ( S I ) 17.1 f 1.5 17.2 f 1.1 KI 28.5 f 1.4 24.0 f 1.5 K2 1.4 0.1 K3 1.7 f 0.1 P (Pi) 2.9 f 0.2 2.6 f 0.4
I
*
-A
P
*
The time dependence of the exchange cross peaks and corresponding diagonal resonances in a series of NOESY spectra recorded with mixing times ranging from 10 to 300 ms at pH 5.8 is shown in Figure 2, together with the best-tit theoretical curves using the simple three-species model of Scheme I. A is the major species, B and C are the two minor species, ki are the rate constants for interconversion between the species, and p is the total spinlattice relaxation rate of the relevant proton, which for simplicity is assumed to be the same for the two histidines of all three species. The time development of magnetization in such a system is described byS dMA/dt
-MA(P + k-l
dMB/dt = - M B ( ~+ kl dMc-/dt
*
+ k-2) + MBkl + Mckz
+ k-3) + MAk-1 + Mck3 = -Mc(p + k2 + k , ) + M ~ k - 2+ M ~ k - 3
(1)
The equilibrium constants between the species are defined as KI = k l / k - l = [Al/[B], K2 = k2/k:2 = [A]/[C], and K3 = k3/k-3 = [B]/[C] = K 2 / K , . The intensity of a given diagonal peak and its associated cross peaks as a function of mixing time t is obtained by solving eq 1 with the magnetization of the species corresponding to the diagonal peak set to 1 in the case of the major species A (and to l / K I or 1/K2 in the case of minor species B or C) and the magnetization of all other species set to 0. The complete set of build-up curves was fitted simultaneously by carrying out successive numerical integration runs under control of a nonlinear least-squares optimization routine, varying the values of the rate constants k l , k-', k2, k-2, and p , and a scale factor. The exact values of k3 and k-3 have a minimal effect on the system as the equilibrium constant K3 is determined by the other two equilibrium constants, so that for simplicity k-3 was set to lo4 s-l. The diagonal and exchange cross peaks of the two histidine residues exhibit the same time dependence, indicating that they are associated with the same dynamical processes. The results of the analysis are summarized in Table I for data collected at pH 5.8 and 7.0. At pH 5.8,91.3% of the zinc finger is in form A, 5.3% in form B,and 3.4% in form C, and there is no significant change in these values at pH 7.0. The main difference between the data at pH 5.8 and 7.0 lies in the rates for the interconversion between species A and C, which are approximately half those at pH 5.8. At pH values above 5.5 the equilibrium constant for the binding of zinc to the peptide is L106M-l so that, under the experimental conditions employed, the concentration of zinc-free peptide is
